A circularity in HTT

The corresponding argument in Lurie (2009) contains a circular dependency. Proposition~7.2.1.14 is the analogue of the lemma above. Its proof uses Lemma~7.2.1.13, which invokes Proposition~6.5.1.20; the proof of Proposition~6.5.1.20 in turn appeals to Proposition~7.2.1.14. Proposition~6.5.1.12 also uses Proposition~7.2.1.14.

There is a related gap in the proof of [Lurie 2009, Proposition 6.5.1.18]: one still has to show that the diagonal of a \(0\)-connected morphism is an effective epimorphism. The proof below establishes the effective-epimorphism criterion independently, and we will use it to supply this step in Theorem 3.22.